# Development of 3D Models of Knits from Multi-Filament Ultra-Strong Yarns for Theoretical Modelling of Air Permeability

^{1}

^{2}

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

- through the pressure drop coefficient, where the overpressure gradient can be determined using the Expression (1):$$\frac{\partial p}{\partial {X}_{i}}={K}_{i}\frac{\rho {V}_{i}^{2}}{2}$$
- through the friction index [14]. In such case, the gradient of additional pressure can be expressed by the Equation (2):$$\frac{\partial p}{\partial {X}_{i}}=\frac{f}{{D}_{h}}\frac{\rho {V}_{i}^{2}}{2}$$
_{h}is the hydraulic diameter in mm. - through the Darcy ratio (3):$$\frac{\partial p}{\partial {X}_{i}}=C\mu Vi$$

**V**(m/s) can be calculated using the Expression (4):

_{h}is the hydraulic diameter of the pore in mm; L is the thickness of the porous material in mm.

^{2}; P is perimeter of the pore in mm.

^{3}) that passes through 1 m

^{2}of textile fabric in 1 second at a certain pressure difference on both sides of the fabric. The air permeability of knitted fabrics was determined in accordance with DSTU ISO 9237: 2003. Ten experimental tests were performed for each sample variant. All measurements were carried out in a standard atmosphere according to Standard EN ISO 139:2005 (20 °C ± 2 °C temperature and 65% ± 4% humidity).

## 3. Results and Discussions

#### 3.1. Modelling of the Knitted Loop

**w**, course spacing

**c**, thickness of the knitted fabric

**M**, yarn diameter

**D**, and the tangent angle at the interlacing point

**γ**, as well as the angle of inclination of the loop legs

**α**(as shown in Figure 9b). These initial data can be obtained experimentally or calculated according to well-known methods, for example, those presented in [21]. For the model of the knitted structure, maximally stretched along wales,

**w = w**and

_{max}**c = c**, while for the model of knitted structure, maximally stretched along courses,

_{min}**w = w**and

_{min}**c = c**. The effective radius

_{max}**R**of a multifilament yarn [31] can be determined by the formula (6):

_{0}_{0}:

_{0}= 0.534 mm.

**w**takes the value

**w**when the yarn is redistributed and pulled from the loops’ heads and feet into the loop legs. In this case w

_{min}_{min}= 4·D

_{0}, id est, the minimum wale spacing is equal to four minimum yarn diameters (because the yarn is in a compressed state). If we accept the assumption of a circular cross-section of the yarn, then the value of the minimum wale spacing, considering Expression (6), will be equal to 2.14 mm, which significantly exceeds the experimentally obtained value for the stretched knitted fabric, which is 1.58 mm (see in Table 1). However, if the bundle of filaments has an elliptical cross-section, and the position of the major and minor axes of the ellipse changes along the axial line of the yarn as shown in Figure 8b, the ratio of the values of the major and minor semiaxes of the ellipse can be selected, at which the area of the ellipse is equal to the area of the circle of the radius determined by Equation (6), and their projection onto the plane of the fabric is equal to the width of the projection of the yarn onto the plane of the fabric, determined experimentally, as shown in Figure 8a.

_{0}, P

_{1}, P

_{2}, P

_{3}, which lie at the intersection of tangents to the axial line drawn at points K, B and T, are shown accordingly in Figure 9a,b. To describe the axes of the ellipse, the following notation is used: P

_{max}is the large, and P

_{min}is the minor axis of the ellipse. The axial line equation is built on the basis of the coordinates of the control points, and those, in turn, can be calculated based on traditional ideas about the geometry of the knitted loop, taking into account certain assumptions.

_{k}, Y

_{k}, Z

_{k}), A (X

_{a}, Y

_{a}, Z

_{a}), B (X

_{b}, Y

_{b}, Z

_{b}), C (X

_{c}, Y

_{c}, Z

_{c}), T (X

_{t}, Y

_{t}, Z

_{t}) can be calculated taking into account the accepted assumptions about the symmetric shape of the loop and the known characteristics of the structure: w, c, P

_{max}, P

_{min}, M. Mathematical expressions for determining the coordinates of the characteristic points of the loop in the selected coordinate system are presented in Table 2.

_{0}P

_{1}) is parallel to the OX axis and the XOY plane (plane of the fabric). In the projection onto the YOZ plane, the segment P

_{0}P

_{1}is expressed to a point that coincides with the point K. The tangent at point T (segment P

_{2}P

_{3}) is located at an angle (180 − α) to the X-axis. It is also parallel to the XOY plane but located on the opposite side of the XOY than the segment P

_{0}P

_{1}. This is clearly seen in the projection onto the XOZ plane (Figure 10b). The segment P

_{1}P

_{2}intersects with the plane of the fabric (XOY) at point B. Point P

_{1}belongs to the tangent line drawn through the point K, and point P

_{2}belongs to the tangent line drawn through the point T. In accordance with the previously accepted assumptions, the projection of the tangent at point B onto the plane of the fabric is located at an angle γ to the OY axis.

#### 3.2. Determination of Air Permeability and Its Simulation

## 4. Conclusions

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 2.**Images of knitted samples: (

**a**) in a relaxed state; (

**b**) after being stretched along wales; (

**c**) after being stretched along courses.

**Figure 3.**(

**a**) Principal arrangement of the cross-sections of individual filaments in the structure of the yarn and (

**b**) in the yarn bent into the loop.

**Figure 4.**Visualization of the knitted fabric structure with detailing at the level of individual filaments, built in the Autodesk AutoCAD software environment.

**Figure 5.**(

**a**) Model of a knitted sample in a free state detailed at the level of yarn; (

**b**) individual filaments; (

**c**) image of the knitted sample in a free state.

**Figure 7.**Graphical results of the analysis of the air flow through a tube with a linear yarn fragment made as multi-fiber (

**a**) and mono-fiber (

**b**).

**Figure 9.**Characteristic points K, B, T (

**a**) and the control points P0, P1, P2, P3 (

**b**) of the spline curve, which describes a quarter of the axial line of the yarn in the knitted loop of single jersey structure.

**Figure 10.**Meso-model and image of the single jersey knitted fabric made from high molecular weight polyethylene yarns at maximum stretching along wales (

**a**) and along courses (

**b**).

**Figure 11.**Illustration of the process of air passing through the knitted fabric under the condition of a pressure drop of 49 Pa: (

**a**) model of the knitted fabric in a free state, (

**b**) model of the knitted fabric in the state of uniaxial stretching along the wales, (

**c**) model of the knitted fabric in the state of uniaxial stretching along the courses.

Stretching Direction | Wale Spacing w, mm | Course Spacing c, mm | Yarn Diameter D, mm | Fabric Thickness M, mm | Loop Length l, mm |
---|---|---|---|---|---|

Before stretching | 1.92 ± 0.1 | 1.85 ± 0.1 | 0.7 ± 0.05 | 0.85 ± 0.05 | 7.95 ± 0.4 |

Stretched along wales | 1.58 ± 0.1 | 2.78 ± 0.15 | 0.75 ± 0.05 | ||

Stretched along courses | 3.46 ± 0.2 | 0.85 ± 0.05 | 0.82 ± 0.05 |

**Table 2.**Mathematical expressions for determining the coordinates of characteristic points in coordinate system of the loop.

Point | Abscissa | Ordinate | Applicata |
---|---|---|---|

K | ${X}_{k}=0$ | ${Y}_{k}=\frac{c+{P}_{mag}}{2}$ | ${Z}_{k}=\frac{M-{P}_{\mathit{min}}{\_}_{z}}{2}$ |

B | ${X}_{b}=\frac{w}{4}+\frac{{P}_{\mathit{min}}}{2}\mathrm{cos}\gamma $ | ${Y}_{b}=\frac{c}{2}+\frac{{D}_{y}}{2}\mathrm{sin}\gamma $ | ${Z}_{b}=0$ |

T | ${X}_{t}=\frac{A}{4}$ | ${Y}_{t}=0$ | ${Z}_{t}=-\frac{M-{P}_{\mathit{min}}}{2}$ |

Sample No | Experimental Value of Air Permeability, dm^{3}/(m^{2}s) | Simulated Value of Air Permeability, dm^{3}/(m^{2}s) | Discrepancy of the Simulated Value, % |
---|---|---|---|

1 | 1617 | 1687 | 0.04 |

2 | 2353 | 2301 | −0.02 |

3 | 1646 | 1645 | 0.00 |

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**MDPI and ACS Style**

Ielina, T.; Halavska, L.; Mikucioniene, D.; Milasius, R.; Bobrova, S.; Dmytryk, O. Development of 3D Models of Knits from Multi-Filament Ultra-Strong Yarns for Theoretical Modelling of Air Permeability. *Materials* **2021**, *14*, 3489.
https://doi.org/10.3390/ma14133489

**AMA Style**

Ielina T, Halavska L, Mikucioniene D, Milasius R, Bobrova S, Dmytryk O. Development of 3D Models of Knits from Multi-Filament Ultra-Strong Yarns for Theoretical Modelling of Air Permeability. *Materials*. 2021; 14(13):3489.
https://doi.org/10.3390/ma14133489

**Chicago/Turabian Style**

Ielina, Tetiana, Liudmyla Halavska, Daiva Mikucioniene, Rimvydas Milasius, Svitlana Bobrova, and Oksana Dmytryk. 2021. "Development of 3D Models of Knits from Multi-Filament Ultra-Strong Yarns for Theoretical Modelling of Air Permeability" *Materials* 14, no. 13: 3489.
https://doi.org/10.3390/ma14133489